Can a woman leading other women in Tarawih prayer recite aloud?
Yes, a woman leading other women in Tarawih prayer is allowed to recite aloud, as long as no non-mahram (other than unmarriageable kin) men can hear her.
What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.
Why was the exact date of Laylat al-Qadr concealed?
The wisdom behind this is to encourage Muslims to strive in worship throughout all the nights of Ramadan, or at least in the last ten nights. By doing so, they will earn the reward of observing Laylat al-Qadr as well as the reward of worshiping on other blessed nights.
What is the virtue of Laylat al-Qadr?
It is enough to know its great status that Allah revealed an entire surah about it—Surah Al-Qadr. Moreover, the Quran was first revealed on this night.
Allah says {what means}:
"We have indeed revealed this (Message) in the Night of Power: And what will explain to thee what the night of power is? The Night of Power is better than a thousand months. Therein come down the angels and the Spirit by Allah's permission, on every errand: Peace!... This until the rise of morn!" [Al-Qadr/1-5]
The Prophet ﷺ taught us to observe I‘tikaf in the last ten nights of Ramadan, seeking Laylat al-Qadr. Whoever worships Allah and revives this night will receive the reward of worshiping for a thousand months—excluding Laylat al-Qadr itself.